A CMC wall (22) with a front surface (21) heated (24) by a working fluid in a gas turbine. A back CMC surface (23) is coated with a layer (42) of a thermally conductive material to accelerate heat transfer in the plane of the CMC wall (22), reducing thermal gradients (32-40) on the back CMC surface (23) caused by cold spots (32) resulting from impingement cooling flows (26). The conductive material (42) may have a coefficient of thermal conductivity at least 10 times greater than that of the CMC material (22), to provide a minimal thickness conductive layer (42). This reduces thermal gradient stresses within the CMC material (22), and minimizes differential thermal expansion stresses between the CMC material (22) and the thin conductive layer (42).